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SumConvergence

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SumConvergence
gives conditions for the sum to be convergent.
SumConvergence
gives conditions for the multiple sum to be convergent.
  • The following options can be given:
Assumptions$Assumptionsassumptions to make about parameters
Direction1direction of summation
MethodAutomaticmethod to use for convergence testing
  • Possible values for Method include:
"IntegralTest"the integral test
"RaabeTest"Raabe's test
"RatioTest"D'Alembert ratio test
"RootTest"Cauchy root test
  • With the default setting Method->Automatic, a number of additional tests specific to different classes of sequences are used.
  • For multiple sums, convergence tests are performed for each independent variable.
Test for convergence of the sum :
Test the convergence of :
Find the condition for convergence of :
Test for convergence of the sum :
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Test the convergence of :
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Find the condition for convergence of :
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Exponential or geometric sums:
Plot the partial sums:
Polynomial exponential sums:
Rational sums:
Convergence picture:
Special functions:
Piecewise functions:
Slowly converging sums in the Abel-Dini scale:
Alternating sums:
Complex-valued sums:
Exponential or geometric series:
Parameter region for convergence:
Power series:
The convergence region for :
Combined series:
Piecewise sums:
Assuming to be complex:
A multivariate sum:
The ratio test typically applies to exponential and hypergeometric terms:
In this case the ratio test is inconclusive:
The root test typically applies to exponential terms:
In this case the root test is inconclusive:
The Raabe test works well for rational functions:
In this case the Raabe test is inconclusive:
The integral test works well on logarithmic terms:
In this case the integral test is inconclusive:
Find the radius of convergence of a power series:
Prove convergence of Ramanujan's formula for :
Sum it:
Convergence properties are not affected by multiplication of constants:
Convergence is not affected by translating arguments:
SumConvergence is automatically called by Sum:
Many conditions generated by Sum are in effect convergence conditions:
With the setting VerifyConvergence->False, typically a regularized value is returned:
SumConvergence is used in sum transforms such as ZTransform:
Conditionally convergent periodic sums:
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